Combinatorial Self-Similarity

Combinatorial (or numerical) self-similarity is an apparently new concept, introduced here in an attempt to explain the similarity of properties of the members of a homologous series that are not (geometrically) self-similar and whence are not (deterministic) fractals. The term is defined in the following steps: a) Select a numerical invariant, ep, characteristic of the member of the series b) Partition this property, ep, into a finite number of parts through a prescribed algorithm c) Members are described so as to be combinatorially self-similar (or to represent a »numerical« fractal) if the limits of the ratios of ep of two successive members at infinite stages of homologation are equal for all parts, and equal the corresponding limit for the total property. In the present work, ep is taken to be the Kekule count, K, when dealing with benzenoid systems and the topological index, Z, (H. Hosoya, Buzz. Chem. Soe. Japan 44 (1971) 2332) when dealing with saturated hydrocarbons. The previously described equivalence relation, l, [S. El-Basil, J. Chem. Soe. Faraday Trans. 89 (1993) 909; J. Mol. Struet. (Theochem) 288 (1993) 67; J. Math. Chem. 14 (1993) 305; J. Mol. Struet. (Theoehem) 313 (1994) 237; J. Chem. Soe. Faraday Trans. 90 (1994) 2201], is used to partition K when the number of terminal hexagons remains constant throughout the series; otherwise the method of Klein and Seitz [D. J. Klein and W.A. Seitz, J. Mol. Struet. (Theochem) 169 (1988) 167] is used. For alkanes, an appropriate recurrence relation is used to partition the Z values. It was found that ep for any homologous series of unbranehed benzenoids, alkanes, Clar graphs, rook broads and King polyominos are all scaled by the golden mean, t = 1,618033989, while homologous series of other types of benzenoids also represent »numerical«fractals, but the characteristic scaling factors depend on the closed form expressions of their K values. In all cases, self-similarities were manifested by expressing the ratios of adjacent t.p\u27s in the form of continued fractions, which in some cases led to exact selfsimilarity but in most cases self-similarity was only approximate